Classical R-matrices, orbit method and finite-gap integration of the Boussinesq hierarchy

We construct an integrable hierarchy of the Boussinesq equation using the Lie-algebraic approach of Holod-Flaschka-Newell-Ratiu. We show that finite-gap Hamiltonian systems of the hierarchy arise on coadjoint orbits in the loop algebra of sl(3) and possess spectral curves from the family of (3, 3N + 1)-curves, N∈N. Separation of variables leads to the Jacobi inversion problem on these curves, which is solved in terms of the corresponding multiply periodic functions. Finally, we explicitly obtain exact finite-gap solutions to the Boussinesq equation, propose a conjecture on reality conditions, and compute solutions for several specific spectral curves with accompanying visualizations.